Citation: Yan Ning, Daowei Lu. A critical point theorem for a class of non-differentiable functionals with applications[J]. AIMS Mathematics, 2020, 5(5): 4466-4481. doi: 10.3934/math.2020287
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The mountain pass theorem of Ambrosetti and Rabinowitz for C1 functions plays an essential role in the area of nonlinear analysis. We are interested in the monograph [22] of Motreanu and Panagiotopoulos in which they established a new version of the mounbtain pass theorem [22,Theorem 3.2] for the functionals f from Banach space X to R∪{+∞} satisfying the following hypothesis:
(Hf):f(x)=Φ(x)+Ψ(x) for all x∈X, where Φ:X→R is locally Lipschitz continuous while Ψ:X→R∪{+∞} is convex, proper, and lower semi-continuous.
We call x∈X is a critical point of f if x solves the following problem:
Φ0(x;z−x)+Ψ(z)−Ψ(x)≥0, ∀ z∈X, | (1.1) |
where Φ0(x;z−x) is the generalized directional derivative of Φ at x in the direction z−x (see [6] for detail).
Recall that f satisfies the (PS)c condition if any sequence {xn}⊂X for which limn→+∞f(xn)=c and
Φ0(xn;z−xn)+Ψ(z)−Ψ(xn)≥−ϵn‖z−xn‖, ∀n∈N,z∈X |
where ϵn→0+, possesses a convergent subsequence.
When (PS)c holds true at any level c we simply write (PS)f in place of (PS)c.
Inequality (1.1) is usually called variational-hemivariational inequality, which has been exploited for mathematically formulating several engineering, besides mechanical questions and extensively studied from many points of view in the latest years [1,22,23].
Variational-hemivariational inequalities can be studied in the framework of a general critical point theory which combines features of the classical convex analysis and of the theory of generalized gradients for locally Lipschitz functions. Such inequalities represent a very general pattern for several kinds of variational problems. Indeed, if Φ∈C1(X,R), the problem (1.1) is reduced to a variational inequality and the relevant critical point theory as well as significant applications are developed in [25]; if Ψ≡0, then (1.1) coincides with the problem treated by Chang in [5] which is called differential equations with discontinuous nonlinearities; differential inclusions (see [12]) and special non-smooth problems with constraints (see [13]) can be considered as special cases of variational-hemivariational inequality. Finally, when both Φ∈C1(X,R) and Ψ≡0, the problem (1.1) becomes the Euler equation Φ′(u)=0 and the theory is classical. For the new results on this topic, see the excellent overview in [4,7,10,11,14,15,17,18,19,26,27].
Chang in [5] established the critical point theory for non-differentiable functionals and represented some applications to partial differential equations with discontinuous nonlinearities. Marano and Motreanu [20] obtained a critical points theorem which extends the variational principle of Ricceri to variational-hemivariational inequalities and semilinear elliptic eigenvalue problems with discontinuous nonlinearities. The critical point theorem in presence of splitting was established by Brˊezis-Nirenberg [3]. Subsequently Livrea, Marano and Motreanu [16] extended it to Motreanu-Panagiotopoulos' setting under the the following structural hypothesis (Hf)′:
(Hf)′: f(x)=Φ(x)+Ψ(x) for all x∈X, where Φ:X→R is locally Lipschitz continuous while Ψ:X→R∪{+∞} is convex, proper, and lower semi-continuous, and Ψ is continuous on any nonempty compact set A⊆X such that supx∈AΨ(x)<+∞.
And they applied the conclusions to an elliptic variational-hemivariational inequality.
Motivatied by the above cited papers, we try to prove a multiplicity theorem of functions f fulfilling the structural hypothesis (Hf), mountain pass geometry and the bounded from below conditions. In Section 2, we will recall some basic definitions and preliminaries. The essential tool used in the proof is a general deformation lemma, which will be set forth in Section 3. Section 4 presents our main result, a new critical point theorem.
In the last section we consider an application to the elliptic variational-hemivariational inequality:
(Pλ): Find u∈Kλ such that for all v∈Kλ,
−∫Ω∇u(x)∇(v−u)(x)dx−∫Ωa(x)u(x)(v−u)(x)dx≤λG0(u;v−u), |
where λ>0, Kλ is convex and closed in H10(Ω), g:R→R is locally bounded and measurable, and the functions G:R→R and G:H10(Ω)→R given by
G(ξ)=∫ξ0−g(t)dt, ∀ ξ∈R, G(u)=∫ΩG(u(x))dx, ∀ u∈H10(Ω), |
respectively, are well defined and locally Lipschitz continuous. Some results of [16] are improved.
Let (X,‖⋅‖) be a reflexive Banach space. Denote by B(x,δ):={z∈X:‖z−x‖<δ} as well as Bδ:=B(0,δ). The symbol [x,z] denote the segment joining x to z, namely [x,z]:={(1−t)x+tz:t∈[0,1]}, and (x,z]:=[x,z]∖{x}. We denote by X∗ the dual space of X, while ⟨⋅,⋅⟩ stands for the duality pairing between X and X∗. A functional φ:X→R∪{+∞} is proper if Dφ={x∈X:φ(x)<∞}≠∅. Functional Φ:X→R is called locally Lipchitz continuous if for every x∈X there exists a neighborhood Vx of x and a constant Lx≥0 such that
|Φ(z)−Φ(w)|≤Lx‖z−w‖, ∀z,w∈Vx. |
Let Φ0(x;z) be the generalized directional derivative of Φ at x along the direction z, i.e.,
Φ0(x;z):=lim supw→x,t→0+Φ(w+tz)−Φ(w)t. |
The generalized gradient of the function Φ at x, denoted by ∂Φ(x), is the set
∂Φ(x):={x∗∈X∗:⟨x∗,z⟩≤Φ0(x;z), ∀z∈X}. |
The mapping z↦Φ0(x;z) is positively homogeneous and sub-additive, thus, due to the Hahn-Banach theorem, the set ∂Φ(x) is nonempty. In the sequel, we state the main properties of the generalized directional derivatives and the generalized gradients:
1) For each x∈X, ∂Φ(x) is a nonempty, convex in addition to weak∗ compact subset of X∗.
2) For each x,z∈X,Φ0(x,z) is upper semicontinuous on X×X.
3) For each x,z∈X, we have Φ0(x;z)=max{⟨x∗,z⟩; x∗∈∂Φ(x)}.
4) If Φ attains a local minimum or maximum at x, then 0∈∂Φ(x).
5) The function mΦ(x)=min{‖x∗‖X∗,x∗∈∂Φ(x)} exists and is lower semi-continuous.
Let Ψ:X→R∪{+∞} be convex, proper, and lower semi-continuous. Set DΨ={x∈X:Φ(x)<+∞}, then Ψ is continuous in int(DΨ) (see [8]). To simplify notation, denote by ∂Ψ(x) the subdifferential of Ψ at the point x∈X in the sense of convex analysis, while D∂Ψ={x∈X:∂Ψ(x)≠∅}. By [8], int(DΨ)=int(D∂Ψ), ∂Ψ(x) is convex and weak∗ closed.
Let f be a function on X satisfying the hypothesis(Hf), a∈R. Define
Ka(f)={x∈X:f(x)=a, x is a critical point of f},fa={x∈X:f(x)≥a},fa={x∈X:f(x)≤a}. |
For every ϵ,r>0, we introduce the set
Fra,ϵ={x∈X:‖x‖≤r+1,and |f(x)−a|≤ϵ}, |
it is easy to see that Fra,ϵ is closed.
In this section we establish a deformation lemma for the functions satisfying the hypothesis (Hf).
Lemma 3.1. Suppose x∈int(DΨ). Then for every xn→x in X and every z∗n∈∂Ψ(xn),n∈N, there exists z∗∈∂Ψ(x) as well as a subsequence {z∗rn} of {z∗n} such that z∗rn⇀z∗ in X∗.
For the proof the reader could refer to [21,Remark 2.1].
Lemma 3.2. Let f be a function satisfying (Hf). Assume that there exist constants ϵ>0,r>0 and a∈R such that Fra,ϵ≠∅,Fra,ϵ⊆int(DΨ), and
inf{‖x∗+z∗‖:x∗∈∂Φ(x),z∗∈∂Ψ(x),x∈Fra,ϵ}>2ϵ. |
Then for every x∈Fra,ϵ, there exists ξx∈X such that
‖ξx‖=1,⟨x∗+z∗,ξx⟩>2ϵ,forall x∗∈∂Φ(x), z∗∈∂Ψ(x). | (3.1) |
Proof. Since inf{‖x∗+z∗‖:x∗∈∂Φ(x), z∗∈∂Ψ(x), x∈Fra,ϵ}>2ϵ, there exists an ϵ0>0 such that for every x∈Fra,ϵ,x∗∈∂Φ(x),z∗∈∂Ψ(x), we have ‖x∗+z∗‖X∗≥2ϵ+ϵ0.
Fix an x∈Fra,ϵ, since ∂Φ(x) and ∂Ψ(x) are nonempty and convex, so is ∂Φ(x)+∂Ψ(x). As X is reflexive, ∂Φ(x) is weak∗ compact and ∂Ψ(x) is weak∗ closed, then ∂Φ(x)+∂Ψ(x) is closed.
Note that 0∉∂Φ(x)+∂Ψ(x). By [2, Corollary 3.20], we have u∗∈∂Φ(x),v∗∈∂Ψ(x) satisfying
Bδ∗∩(∂Φ(x)+∂Ψ(x))=∅, where δ∗=‖u∗+v∗‖X∗>0. |
Now the Hahn-Banach theorem provides a point ξx∈X with ‖ξx‖=1 and whenever x∗∈∂Φ(x),z∗∈∂Ψ(x),
⟨x∗+z∗,ξx⟩≥⟨w∗,ξx⟩, ∀ w∗∈Bδ∗. |
Since ‖u∗+v∗‖X∗=‖u∗+v∗‖X∗‖ξx‖=max{⟨w∗,ξx⟩,w∗∈¯Bδ∗}, the above inequality and Lemma 3.1 lead to
⟨x∗+z∗,ξx⟩≥‖u∗+v∗‖X∗≥2ϵ+ϵ0>2ϵ, ∀ x∗∈∂Φ(x), z∗∈∂Ψ(x). |
The proof is completed.
Lemma 3.3 Under the conditions of Lemma 3.2, for every x∈Fra,ϵ, there exists a δx>0 such that
⟨x∗+z∗,ξx⟩>2ϵ, ∀ x∗∈∂Φ(x′),z∗∈∂Ψ(x″), ∀ x′, x″∈B(x,δx), | (3.2) |
where ξx is given by Lemma 3.2.
Proof. If the conclusion were false, then we could find x∈Fra,ϵ,{x′n},{x″n}⊆X and {x∗n},{z∗n}⊆X∗ such that
x′n→x,x∗n∈∂Φ(x′n), ∀ n∈N; | (3.3) |
x″n→x,z∗n∈∂Ψ(x″n), ∀ n∈N; | (3.4) |
⟨x∗n+z∗n,ξx⟩≤2ϵ, ∀ n∈N. | (3.5) |
Due to the reflexivity of X and (3.3), Proposition 2.1.2 of [22] yields x∗∈X∗ such that x∗n⇀x∗ in X∗, where a subsequence is considered when necessary, while Proposition 2.1.5 of [22] forces x∗⊆∂Φ(x). Since x∈Fra,ϵ⊆int(DΨ)=int(D∂Ψ), combining (3.4) with Lemma 3.1, we obtain, up to subsequences, z∗n⇀z∗ for some z∗∈∂Ψ(x). Now from (3.5) it follows, as n→+∞, ⟨x∗+z∗,ξx⟩≤2ϵ. However, this contradicts (3.1).
Theorem 3.4 Let f be a function satisfying (Hf), assume that there exist constants ϵ>0,r>0 and a∈R such that Fra,ϵ≠∅,Fra,ϵ⊆int(DΨ), and
inf{‖x∗+z∗‖:x∗∈∂Φ(x),z∗∈∂Ψ(x),x∈Fra,ϵ}>2ϵ. |
Then there exists a continuous mapping η:X→X with the following properties:
(1)η:X →X isahomeomorphism;(2)η(x)=x whenever |f(x)−a|≥2ϵ;(3)‖η(x)−x‖≤1, ∀x∈X;(4)f(η(x))≤f(x), ∀x∈X;(5)η(DΨ)⊆DΨ;(6)f(η(x))≤a−ϵ, ∀ x∈X provided ‖x‖≤r and f(x)≤a+ϵ. |
Proof. The family of balls B={B(x,δx):x∈Fra,ϵ} constructed through Lemma 3.3 represents an open covering of Fra,ϵ, and the assumptions ensure that Fra,ϵ is a nonempty para-compact set because it is closed. So B possesses an open locally finite refinement V={Vi;i∈I}. Moreover, to each i∈I there corresponds ξi∈X such that ‖ξi‖=1 as well as
⟨x∗+z∗,ξi⟩>2ϵ, ∀ x∗∈∂Φ(x′),z∗∈∂Ψ(x″), ∀ x′,x″∈Vi. | (3.6) |
Shrink V to an open locally finite covering W={Wi;i∈I} fulfilling for every i∈I, Wi⊆Vi ([9,Theorems Ⅷ 2.2 and Theorems Ⅶ 6.1]) with Wi is convex, and f|Wi(⋅) is Lipschitz continuous satisfying
a−2ϵ<f(x)<a+2ϵ, ∀ x∈Wi. | (3.7) |
Set
W=⋃i∈IWi,di(x)=d(x,X∖Wi),ρi(x)=di(x)∑j∈Idj(x), ∀ x∈W, i∈I,Θ(x)={∑i∈Iρi(x)ξi, if x∈W,0, otherwise,l(x)=d(x,X∖W)d(x,X∖W)+d(x,Fra,ϵ), ∀ x∈X,V(x)=l(x)Θ(x), ∀ x∈X,V(x)=l(x)Θ(x), ∀ x∈X. |
We observe that V:X→X is locally Lipschitz continuous and
‖V(x)‖≤1, ∀ x∈X. | (3.8) |
The existence-uniqueness theorem for ordinary differential equations provides a mapping σ∈C0(R×X,X) such that
dσ(t,x)dt=−V(σ(t,x)), ∀ (t,x)∈R×X, σ(0,x)=x. |
We claim that
for every x∈X, the function t↦f(σ(t,x)) is non-increasing on R. | (3.9) |
In fact, if x∈X∖W, V(x)=0 and thus σ(⋅,x) is constant and (3.9) holds true. In the case x∈W, we start by noting that σ(R,x)⊆W. Indeed setting T=sup{t>0:σ((−t,t),x)⊆W}, assume by contradiction that T<+∞. Hence we have Wx=σ(T,x)=limt→T−σ(t,x)∈∂W. Then the Cauchy problem
d˜σ(t)dt=−V(˜σ(t)), ∀ t∈R, ˜σ(T)=Wx |
admits the constant solution ˜σ(⋅)≡Wx, as does t↦σ(t,x) for t≤T, which is against the uniqueness of solutions.
Fixing t∈R, we know that σ(t,x)∈W and due to the local finiteness of W, the set J={i∈I:σ(t,x)∈Wi} is finite. It follows that ˜W=∩i∈JWi is a convex, open neighborhood of σ(t,x), and there exists δ>0 such that
σ((t−δ,t+δ),x)⊆˜W, and σ((t−δ,t+δ),x)⋂(⋃i∈I∖JWi)=∅. | (3.10) |
For arbitrary t′,t″∈(t−δ,t+δ) with t′<t″. Lebourg's mean value theorem provides y∈(σ(t′,x),σ(t″,x)), x∗∈∂Φ(y), z∗∈∂Ψ(y) satisfying
f(σ(t′,x))−f(σ(t″,x))=⟨x∗+z∗,σ(t″,x)−σ(t′,x)⟩=−∫t″t′⟨x∗+z∗,V(σ(τ,x))⟩dτ<−2ϵ∫t″t′l(σ(τ,x))∑j∈Jρj(σ(τ,x))dτ=−2ϵ∫t″t′l(σ(τ,x))dτ, |
where (3.6) and (3.10) have been used. Given p,q∈[t,t+1] with p<q, a standard compactness argument and the above estimate enable us to find t1,t2,...,ts∈[t,t+1] with p=t1<t2<...<ts=q such that
f(σ(ti,x))−f(σ(ti−1,x))<−2ϵ∫titi−1l(σ(τ,x))dτ, |
for all i=1,2,...,s. It turns out that
f(σ(q,x))−f(σ(p,x))=s∑i=1[f(σ(ti,x))−f(σ(ti−1,x))]<−2ϵ∫qpl(σ(τ,x))dτ<0, | (3.11) |
which establish (3.9). Now we define
η(x)=σ(1,x), ∀ x∈X. |
From the general theory of ordinary differential equations it is well known that η:X→X is a homeomorphism.
Since (3.7) renders {x∈X:|f(x)−a|≥2ϵ}⊆X∖W when x∈X∖W, V(x)=0, thus σ(⋅,x) is constant, and η(x)=σ(1,x)=σ(0,x)=x. We then deduce property (2).
From (3.8), for all x∈X, we have
‖η(x)−x‖=‖σ(1,x)−σ(0,x)‖=‖∫10V(σ(τ,x))dτ‖≤∫10‖V(σ(τ,x))‖dτ≤1, |
i.e., property (3) holds true.
Since f(η(x))=f(σ(1,x))≤f(σ(0,x))=f(x), for all x∈X, the property (4) holds true.
For every x∈DΨ, there is f(x)<+∞. Since (4) holds, f(η(x))≤f(x)<+∞, so η(x)∈DΨ. i.e. η(DΨ)⊆DΨ, (5) holds true.
In order to prove (6), let x∈X with ‖x‖≤r and f(x)≤a+ϵ. If f(x)≤a−ϵ, (6) follows from (4) immediately. In case a−ϵ<f(x)≤a+ϵ, we argue by contradiction. Suppose
a−ϵ<f(η(x))=f(σ(1,x))≤f(σ(t,x))≤f(x)≤a+ϵ, ∀ t∈[0,1]. | (3.12) |
In addition, through (3.8), for all t∈[0,1] we have
‖σ(t,x)‖≤‖x‖+‖σ(t,x)−x‖≤r+‖∫t0dσ(τ,x)dτdτ‖≤r+∫t0‖V(σ(τ,x))‖dτ≤r+1. |
Consequently, σ([0,1],x)⊆Fra,ϵ, which forces l(σ(⋅,x))|[0,1]≡1. Then (3.11) with p=0 and q=1 reads as
f(η(x))−f(x)<−2ϵ. | (3.13) |
Combining (3.12) and (3.13) gives
a−ϵ<f(η(x))<f(x)−2ϵ≤a−ϵ, |
which is a contradiction.
Fix v0, v1∈DΨ. Consider the following set of paths
Γ={γ∈C0([0,1],X):γ(0)=v0, γ(1)=v1}, | (4.1) |
and a function f:X→R∪{+∞} which verifies hypothesis (Hf). Set
c=infγ∈Γsupt∈[0,1]f(γ(t)). | (4.2) |
Theorem 4.1 Suppose f:X→R∪{+∞} satisfies (Hf) and (PS)f. Assume in addition that
(i1)f is bounded below and coercive, and put α=infx∈Xf(x);
(i2)α<max{f(v0),f(v1)}≤c, v0≠v1 and for every γ∈Γ, there exists t∈(0,1) such that f(γ(t))≥max{f(v0),f(v1)};
(i3) for every a∈R, there exist r>0 and ϵ0>0 such that Fra,ϵ0⊆int(DΨ).
Then the function f possesses at least two critical points.
Proof. Since f is coercive, for every x∈X with c−1≤f(x)≤c+1, there exists a constant k such that
‖x‖≤k. | (4.3) |
From (i3), for c and k>0, there is ϵ0>0 such that
Fkc,ϵ0⊆int(DΨ). | (4.4) |
Without loss of generality, we assume ϵ0<1. Let
cϵ=infγ∈Γsupt∈[0,1][f(γ(t))+ϵd(t)], | (4.5) |
where 0<ϵ<12ϵ0 is arbitrary, and d(t)=min{t,1−t}, t∈[0,1].
From (i2), we can easily verify that
c≤cϵ<c+ϵ. |
Since for every γ∈Γ, there exists t0∈(0,1) such that f(γ(t0))≥max{f(v0),f(v1)}, one has
supt∈[0,1](f(γ(t))+ϵd(t))≥f(γ(t0))+ϵd(t0)>f(γ(t0))≥max{f(v0),f(v1)}, |
thus cϵ>max{f(v0),f(v1)} for every ϵ∈(0,12ϵ0).
We claim that for every ϵ∈(0,12ϵ0) satisfying that ϵ<12(cϵ−max{f(v0),f(v1)}) there holds
inf{‖x∗+z∗‖:x∗∈∂Φ(x), z∗∈∂Ψ(x), x∈Fkcϵ,ϵ}≤2ϵ. | (4.6) |
Due to the definition of ˆϵ, for every ϵ∈(0,ˆϵ), we have
c−ϵ0<c−ϵ<cϵ−ϵ≤f(x)≤cϵ+ϵ<c+2ϵ<c+ϵ0 |
and ‖x‖≤k. It is straightforward to verify that Fkcϵ,ϵ⊆int(DΨ).
To show (4.6), we argue by contradiction. If it was not true, then we would find ϵ∈(0,ˆϵ) for which Theorem 3.4 can be applied with a=cϵ and r=k. So there would exist a continuous mapping η:X→X with the properties (1)–(6) formulated in Theorem 3.4. By the definition of ˆϵ, there is γϵ∈Γ such that
cϵ≤supt∈[0,1][f(γϵ(t))+ϵd(t)]<cϵ+ϵ, |
which easy to verify that
cϵ−ϵ<supt∈[0,1]f(γϵ(t))<cϵ+ϵ, |
and use the definition of ˆϵ again, it is straightforward to verify that η(γϵ(⋅))∈Γ for every ϵ. Hence, in view of (4.5), we may consider a sequence {sϵ} in [0,1] such that
cϵ−ϵ<f(η(γϵ(sϵ))),cϵ−ϵ<f(γϵ(sϵ))<cϵ+ϵ. | (4.7) |
Since cϵ+ϵ<c+2ϵ<c+1 and cϵ−ϵ>c−ϵ>c−1, it implies that ‖γϵ(sϵ)‖≤k.
Exploiting (4.7) and property (6), we achieve the contradiction
cϵ−ϵ<f(η(γϵ(sϵ)))≤cϵ−ϵ. |
thereby (4.6) holds true.
By virtue of (4.6), for every x∈X and all n∈N sufficiently large, there exists xn∈Fkc1n,1n, x∗n∈∂Φ(xn),z∗n∈∂Ψ(xn) such that
‖x∗n+z∗n‖<3n, |
and
c−1n<c1n−1n≤f(xn)≤c1n+1n<c+2n. |
This guarantees that
‖xn‖≤k+1,c−1n<f(xn)<c+2n, |
and
Φ0(xn;x−xn)+Ψ(x)−Ψ(xn)≥⟨x∗n+z∗n,x−xn⟩≥−‖x∗n+z∗n‖‖x−xn‖>−3n‖x−xn‖. |
Since f satisfies the (PS)f condition, there is an ¯x∈X such that xn→¯x in X, where a subsequence is considered when necessary. At this point, ¯x is a critical point of f, and ¯x∈Kc(f).
Next we prove that f possesses a global minimum point x0∈X. Since by (i1) and the condition (PS)f, each minimizing sequence for f possesses a convergent subsequence (see[16]), the function f must attain its minimum at some point x0∈X.
Due to f(x0)=α<c=f(ˉx), x0≠ˉx, which completes the proof.
Remark 4.2 By the above proof, one can find that under the conditions of Theorem 4.1, when a≥α, there exist r>0, ϵ>0 such that Fra,ϵ≠∅. By the coercivity of f, if Ψ is convex and continuous, the condition (i3) obviously holds.
In this section we use Theorem 4.1 to discuss an elliptic variational-hemivariational inequality in the sense of Panagiotopoulos [24].
Let Ω be a nonempty, bounded, open subset of RN (N≥3) with smooth boundary ∂Ω. Denote by H10(Ω) the usual Sobolev space with norm
‖u‖=(∫Ω|∇u(x)|2dx)12. |
It's well known that for p∈[1,2∗], 2∗=2N/(N−2), there exists a positive constant Cp such that
‖u‖Lp(Ω)≤Cp‖u‖, u∈H10(Ω). | (5.1) |
Given a function a∈L∞(Ω) satisfying a(x)≥0 for a.e. x∈Ω. Let
β=essinfx∈Ωa(x)≥0. |
If g:R→R satisfies the condition
(g1) g is locally bounded and measurable.
Then the functions G:R→R and G:H10(Ω)→R given by
G(ξ)=∫ξ0−g(t)dt, ∀ ξ∈R,G(u)=∫ΩG(u(x))dx, ∀ u∈H10(Ω), |
respectively, are well defined and locally Lipschitz continuous. So it makes sense to consider their generalized directional derivatives G0 and G0. On account of [22,formula(9),P.84] one has
G0(u;v)≤∫ΩG0(u(x);v(x))dx, u, v∈H10(Ω). | (5.2) |
We will further assume
(g2) limt→0g(t)t=0;
(g3) lim sup|t|→+∞g(t)t≤0;
(g4) there exists a ξ0∈R such that G(ξ0)<0.
Through (g3) for every ϵ>0 there exists a constant r>0 such that
g(t)≤ϵt, for all |t|≥r. | (5.3) |
Since g is locally bounded, we also have
M=supt∈[−r,r]|g(t)|<+∞. | (5.4) |
Let λ>0, μ(Ω) be the Lebesgue measure of Ω. Define
rλ=√4λ+2Mrμ(Ω). |
A set Kλ⊆H10(Ω) is called of type (Kgλ) provided
(Kgλ):Kλ is convex and closed in H10(Ω). Moreover, ¯Brλ⊆Kλ.
Given λ>0 and Kλ satisfying (Kgλ), consider the elliptic variational-hemivariational inequality problems:
(Pλ): Find u∈Kλ such that for all v∈Kλ,
−∫Ω∇u(x)∇(v−u)(x)dx−∫Ωa(x)u(x)(v−u)(x)dx≤λG0(u;v−u). |
Due to (5.2), any solution u of (Pλ) also fulfills the inequality
−∫Ω∇u(x)∇(v−u)(x)dx−∫Ωa(x)u(x)(v−u)(x)dx |
≤λ∫ΩG0(u(x);(v−u)(x))dx, for all v∈Kλ. |
If g is continuous and Kλ=H10(Ω), the function u∈H10(Ω) turns out a weak solution to the Dirichlet problem
{−Δu+a(x)u=λg(u),inΩ,u=0,on∂Ω, |
which has been previously investigated in [3,14] under more restrictive conditions.
Theorem 5.1 Suppose (g1)−(g4) hold true. Then, for every λ sufficiently large, problem (Pλ) possesses at least two solutions.
Proof. Let X=H10(Ω), p∈(2,2∗). Define a functional f(u)=Φ(u)+Ψ(u) on X as follows:
Φ(u)=12∫Ω(|∇u(x)|2+a(x)u(x)2)dx+λG(u) |
as well as
Ψ(u)={0, if u∈Kλ,+∞, otherwise, |
where λ>0 and Kλ⊆H10(Ω) is of type (Kgλ). Owing to (g1) the function Φ:X→R turns out locally Lipschitz continuous. Consequently, f satisfies condition (Hf).
We shall prove that f is bounded from below and coercive.
By (5.3) and (5.4), one has
∫ξ0g(t)dt≤Mr+ϵ2ξ2, ∀ ξ∈R. | (5.5) |
Which clearly implies
G(u)≥−Mrμ(Ω)−ϵ2‖u‖2L2(Ω), ∀ u∈X. |
Then we obtain
f(u)≥Φ(u)≥12‖u‖2+β2‖u‖2L2(Ω)−λϵ2‖u‖2L2(Ω)−λMrμ(Ω)=ϵ2‖u‖2+12(β−ϵλ)‖u‖2L2(Ω)−λMrμ(Ω). |
Setting ϵ∈(0,βλ), then we have
f(u)≥12‖u‖2−λMrμ(Ω), ∀ u∈X, | (5.6) |
which shows the claim.
Let us next show that the function f satisfies condition (PS)f. Pick a sequence {un}⊆X such that {f(un)} is bounded and
Φ0(un;v−un)+Ψ(v)−Ψ(un)≥−ϵn‖v−un‖. | (5.7) |
for all n∈N,v∈X, where ϵn→0+.
By (5.7) one evidently has {un}⊆Kλ, and {f(un)} is bounded. Since f is coercive, the sequence {un} turns out bounded. Passing to a subsequence if necessary, we suppose un⇀u in X and un→u in L2(Ω). The point u belongs to Kλ because this set is weakly closed.
Exploiting (5.7) with v=u, we then get
∫Ω∇un(x)∇(u−un)(x)dx+∫Ωa(x)un(x)(u−un)(x)dx+λG0(un;u−un)≥−ϵn‖u−un‖, | (5.8) |
for all n∈N.
From un⇀u in X it follows
limn→+∞∫Ωa(x)un(x)(u−un)(x)dx=0. | (5.9) |
The upper semi-continuity of G0 on L2(Ω)×L2(Ω) forces
lim supn→+∞G0(un;u−un)≤G0(u;0)=0. | (5.10) |
Taking account of (5.9), (5.10) besides {‖u−un‖} is bounded, and letting n→+∞, inequality (5.8) yields
lim supn→+∞∫Ω|∇un(x)|2dx≤∫Ω∇|u(x)|2dx. |
Hence, thanks to [2,Proposition Ⅲ.3], un→u in X. i.e., (PS)a holds.
By (g4), we can construct an u0∈X such that G(u0)<0. Moreover, u0∈¯Brλ for any λ≥14‖u0‖2. Therefore, infu∈Xf(u)≤f(u0)<0 provided
λ>max{14‖u0‖2,−12G(u0)∫Ω(|∇u0(x)|2+a(x)u0(x)2)dx}, |
while f(0)=λG(0)=0.
Our next objective is to verify (i1). From (g2), there exists σ∈(0,r) such that
∫|u(x)|<σ[∫u(x)0g(t)dt]dx≤ϵ2∫Ω|u(x)|2dx. | (5.11) |
Due to (5.5), one has
G(ξ)≥−Mr−ϵ2ξ2≥−(Mrσp+ϵ2σp−2)|ξ|p, |
provided |ξ|≥σ.
The Sobolev embedding theorem gives
∫|u(x)|≥σG(u(x))dx≥−(Mrσp+ϵ2σp−2)‖u‖pLp(Ω)≥−C∗‖u‖p, | (5.12) |
where C∗=(Mrσp+ϵ2σp−2)Cpp. Then by (5.11) (5.12) and (5.1) we get
G(u)=∫|u(x)|<σ[∫u(x)0−g(t)dt]dx+∫|u(x)|≥σG(u(x))dx≥−ϵ2C22‖u‖2−C∗‖u‖p=−‖u‖2(ϵ2C22+C∗‖u‖p−2),∀u∈X. | (5.13) |
Let us next prove that for a suitable constant θ>0,
∫Ω(|∇u(x)|2+a(x)u(x)2)dx≥θ∫Ω|∇u(x)|2dx, ∀ u∈X. | (5.14) |
Indeed, if it's not true, there exists a sequence {un}⊆X enjoying the properties
‖un‖=1, n∈N, |
∫Ω(|∇un(x)|2+a(x)un(x)2)dx<1n, ∀ n∈N. | (5.15) |
Passing to a subsequence if necessary, we may suppose un⇀u in X as well as un→u in L2(Ω). Thus, letting n→+∞ in (5.15) yields
∫Ω(|∇u(x)|2+a(x)u(x)2)dx≤0. | (5.16) |
Using the sobolev embedding theorem and β=essinfx∈Ωa(x)≥0 we obtain
(1C22+β)‖u‖2L2(Ω)≤∫Ω(|∇u(x)|2+a(x)u(x)2)dx. | (5.17) |
Gathering (5.16) and (5.17) together, leads to u=0. By (5.15) this forces un→0 in X, against to ‖un‖=1,∀n∈N.
Combining (5.14) with (5.13), provides
f(u)≥‖u‖2(θ2−λ(ϵ2C22+C∗‖u‖p−2)), ∀ u∈X. | (5.18) |
Pick ϵ>0 and R∈(0,12‖u0‖) sufficiently small such that
θ2−λ(ϵ2C22+C∗Rp−2)>0. |
Then by (5.18) we have
f(u)≥0, ∀ u∈¯BR. | (5.19) |
Furthermore, it is easy to prove that R<12‖u0‖<rλ.
Now, let v0=0, v1=u0. Define
Γ={γ∈C0([0,1],X):γ(0)=v0, γ(1)=v1},c=infγ∈Γsupt∈[0,1]f(γ(t)). |
Thanks to (5.19) and the definition of c, one has
c≥0=max{f(v0), f(v1)}, |
and for every γ∈Γ, there exists a t∈(0,1) such that γ(t)∈X and ‖γ(t)‖=R. Then by (5.19) again, we obtain f(γ(t))≥0. Hence hypothesis (i1) of Theorem 4.1 is fulfilled.
Finally, let us prove that (i3) holds. Since f is bounded below, put α=infx∈Xf(x), then α<0≤c. For every a≥α suppose that a<λ, then there exist r>0 and ϵ0>0 such that
Fra,ϵ0⊆int(DΨ). | (5.20) |
Indeed, there is ϵ0>0 such that a+ϵ0≤λ<2λ.
Inequality (5.6) ensures that
{u∈X:f(u)≤a+ϵ0}⊆{u∈X:f(u)≤λ}⫋{u∈X:‖u‖<rλ}⊆¯Brλ⊆DΨ. |
So we immediately have {u∈X:f(u)≤a+ϵ0}⊆int(DΨ).
Since f is coercive, there exists r>0 such that every u∈X satisfies a−ϵ0≤f(u)≤a+ϵ0, and ‖u‖≤r+1, which leads to (5.20), i.e., condition (i3) holds true.
We are now in a position to apply Theorem 4.1. By this theorem, there exist at least two points u1,u2∈X such that
Φ0(ui;v−ui)+Ψ(v)−Ψ(ui)≥0, ∀ v∈X, i=1,2. |
The choice of Ψ gives both ui∈Kλ and Φ0(ui;v−ui)≥0, v∈Kλ, i=1,2. Namely, u1,u2 are solutions to the problem (Pλ).
Example 5.2 The aim of this example is to exhibit a nontrivial case of set in H10(Ω) of type (Kgλ). Let h:H10(Ω)→R be a weakly continuous and convex function. For ˉr>0 fixed, λ>0, put
ˉrλ=√4λ+2Mˉrμ(Ω), |
with the same notation as before. The ball ˉB(0,ˉrλ) is a weakly compact subset of H10(Ω), since h is weakly continuous, there exists u0∈ˉB(0,ˉrλ) such that
γ=maxu∈ˉB(0,ˉrλ)h(u)=h(u0), |
i.e., hˉB(0,ˉrλ) admits a global maximum. Then the set
Kλ:={u∈H10(Ω) : h(u)≤γ+1} |
is a subset of H10(Ω) of type (Kgλ).
Example 5.3 There exist functionals satisfying the conditions of Theorem 5.1. For example
g(t)={|t|(1−e−t2),|t|≤1,t(e−t2−1), |t|>1. |
The authors are grateful to the anonymous referees for their careful reading and helpful comments, which greatly improve the manuscript. The work was supported by the NSF of Shandong Province (No. ZR2018PA006, ZR2017PA001) and the NSF of China (No. 11901240).
The authors declare no conflict of interest.
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